916 resultados para Discrete Mathematics and Combinatorics
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For a polish space M and a Banach space E let B1 (M, E) be the space of first Baire class functions from M to E, endowed with the pointwise weak topology. We study the compact subsets of B1 (M, E) and show that the fundamental results proved by Rosenthal, Bourgain, Fremlin, Talagrand and Godefroy, in case E = R, also hold true in the general case. For instance: a subset of B1 (M, E) is compact iff it is sequentially (resp. countably) compact, the convex hull of a compact bounded subset of B1 (M, E) is relatively compact, etc. We also show that our class includes Gulko compact. In the second part of the paper we examine under which conditions a bounded linear operator T : X ∗ → Y so that T |BX ∗ : (BX ∗ , w∗ ) → Y is a Baire-1 function, is a pointwise limit of a sequence (Tn ) of operators with T |BX ∗ : (BX ∗ , w∗ ) → (Y, · ) continuous for all n ∈ N. Our results in this case are connected with classical results of Choquet, Odell and Rosenthal.
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We prove that in some classes of optimization problems, like lower semicontinuous functions which are bounded from below, lower semi-continuous or continuous functions which are bounded below by a coercive function and quasi-convex continuous functions with the topology of the uniform convergence, the complement of the set of well-posed problems is σ-porous. These results are obtained as realization of a theorem extending a variational principle of Ioffe-Zaslavski.
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* The authors thank the “Swiss National Science Foundation” for its support.
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BOOK REVIEWS Multibody System Mechanics: Modelling, Stability, Control, and Ro- bustness, by V. A. Konoplev and A. Cheremensky, Mathematics and its Appli- cations Vol. 1, Union of Bulgarian Mathematicians, Sofia, 2001, XXII + 288 pp., $ 65.00, ISBN 954-8880-09-01
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Partially supported by Sapientia Foundation.
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∗ Research partially supported by INTAS grant 97-1644
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This article provides necessary and sufficient conditions for both of the Diophantine equations X^2 − DY^2 = m1 and x^2 − Dy^2 = m2 to have primitive solutions when m1 , m2 ∈ Z, and D ∈ N is not a perfect square. This is given in terms of the ideal theory of the underlying real quadratic order Z[√D].
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*This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003
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We discuss functions f : X × Y → Z such that sets of the form f (A × B) have non-empty interiors provided that A and B are non-empty sets of second category and have the Baire property.
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∗ The first and third author were partially supported by National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under grant MM-701/97.
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∗ Supported by the Serbian Scientific Foundation, grant No 04M01
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∗ This work was partially supported by the National Foundation for Scientific Researches at the Bulgarian Ministry of Education and Science under contract no. MM-427/94.
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∗ The first named author’s research was partially supported by GAUK grant no. 350, partially by the Italian CNR. Both supports are gratefully acknowledged. The second author was supported by funds of Italian Ministery of University and by funds of the University of Trieste (40% and 60%).
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Sufficient conditions for the existence of bounded solutions of singularly perturbed impulsive differential equations are obtained. For this purpose integral manifolds are used.
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∗ The final version of this paper was sent to the editor when the author was supported by an ARC Small Grant of Dr. E. Tarafdar.